The $L^\infty$ algebra governing deformation of coisotropic submanifolds in symplectic manifolds is known explicitly in terms of geometric operators, after Oh and Park. We explore a simple example of this deformation problem, particularly the geometry of the canonical foliation. We find that the obstruction theory of the $L^\infty$ algebra makes a fine distinction, of whether the defining parameter is a Liouville number or not.
In this talk we will try to show the hidden geometry (symplectic and Poisson) in what we call b-manifolds. These manifolds were initially considered by Nest and Tsygan while studying formal deformations of symplectic manifolds with boundary and also by Melrose in the context of differential calculus and differential operators of manifolds with boundary. Symplectic b-manifolds lie between the symplectic and Poisson world. In particular, it is possible to prove local and semiglobal normal forms via b-de Rham theory. We will present these results and try to give constructive examples. This talk is based on joink work with Victor Guillemin and Ana Rita Pires.
In this talk I will recall the definition and properties of gaussian maps for curves. I will then concentrate on the second gaussian map, which is related to the second fundamental form of the moduli space of curves. I will finally talk about a recent result in collaboration with A. Calabri and R. Miranda, to the effect that the second gaussian map has maximal rank for a general curve of genus $g$.
We prove a conjecture of B.Shoikhet, relating the (standard) deformation quantization of the symmetric algebra of a finite-dimensional vector field (over a closed field $k$ containing $C$) of Kontsevich, and a less standard one in terms of generators and relations. The key-point lies in the fact that Koszul duality is “preserved” (in a suitable sense) by deformation quantization.
We give an elementary derivation of the Montgomery phase formula for the motion of an Euler top, using only basic facts about the Euler equation and parallel transport on the 2-sphere (whose holonomy is seen to be responsible for the geometric phase). We also give an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point.
Cylindrical contact homology is a powerful invariant of contact structures introduced by Eliashberg, Givental and Hofer in the bigger setup of Symplectic Field Theory. Although it resembles a Floer homology for the action functional induced by the contact form, it has several differences to the Floer homology of monotone symplectic manifolds. Indeed, it is not defined for any contact form and Eliashberg, Givental and Hofer proved that it is well defined and an invariant of the contact structure under some rather restrictive hypotheses on the contact form. We prove an extension of their result to even contact forms, that is, contact forms whose Reeb flow has no periodic orbit of odd degree. A large class of examples of these forms is given by toric contact manifolds whose cylindrical contact homology can be combinatorially computed in terms of the associated momentum cone. This is joint work with Miguel Abreu.
The generalized Segal-Bargmann transform is a unitary map from $L^2$ of a compact group onto a certain $L^2$ space of holomorphic functions on the associated complex group. On the compact group side, one can perform Schrodinger-style quantization of symbols with polynomial behavior in the momentum variables, obtaining differential operators on the compact group. I will show how such operators, when conjugated by the Segal-Bargmann transform, can be expressed as Toeplitz operators. This shows the relationship between Schrodinger and Berezin-Toeplitz quantization in this case.
Inner, Lie and exterior derivatives acting on differential forms are referred to as Cartan calculus operators. In this talk I present a generalization of these notions suitable for bicovariant calculi on quantum groups. Applications to algebraic models for equivariant cohomology of quantum groups will be also briefly discussed. This is a joint work with C. Pagani and A. Zampini.
Narasimhan and Ramanan used the $(0,1)$-stable bundles to define the Hecke curves on the moduli spaces of stable bundles over an algebraic curve. The Hecke curves have been used to study the geometry of the moduli spaces with fixed determinant. In particular, to describe minimal rational curves on the moduli space. Also to describe singular spectral curves and to study the moduli spaces of stable maps of rational curves. In this talk I will present some of the results of my student, Osbaldo Mata, on the generalization of the Hecke curves.
I will describe the structure of the category whose morphisms are riemann surfaces endowed with a principal flat connection for a fixed compact connected structure group. We will relate this space to a suitable stabilization (in genus) of the universal moduli space of riemann surfaces endowed with a holomorphic bundle. The proof involves classical results of Atiyah-Bott along with results of Madsen-Cohen in relation to the parametrized Mumford conjecture. This is joint work with Ralph Cohen and Soren Galatius.
In the first part of the talk I will describe the vortex equations and some of its simplest moduli space of solutions. In the second part I will describe the natural $L^2$-metric on these moduli spaces and explain how, in certain cases, one can explicitly compute its Kahler class and total volume. As a by-product this leads to conjectural formulae for the volume of the space of holomorphic maps of fixed degree from a compact Riemann surface to projective space.
I will explain how the isoperimetric quotient, seen as a functional on sections of a strictly convex polyhedral cone, is related to constant scalar curvature Sasaki toric metric. I will point out the existence of two non-equivalent constant scalar curvature Sasaki toric metrics compatible with the same contact toric structure on $S^2\times S^3$.
A symplectic manifold is a manifold endowed with a non-degenerate closed 2-form (the symplectic form). A Lagrangian is a submanifold of middle dimension on which the symplectic form vanishes. Lagrangians and Hamiltonian diffeomorphisms (diffeomorphisms induced by the flow of certain vector fields) have been extensively studied in different contexts. In this talk, I will show that a Hamiltonian diffeomorphism of a symplectic manifold which preserves a (weakly exact) Lagrangian acts trivially on its homology. The proof is algebraic and relies on standard tools of symplectic geometry (Floer homology and Seidel's morphism) whose construction will be sketched.
If a smooth Riemannian metric on $2$-sphere has curvature sufficiently close to 1 then all closed geodesics are either (i) short and simple or (ii) long and with many self-intersections (Ballmann 1983). We extend this type of result for $r$-reversible Finsler metrics with flag curvature in the interval $(A(r),1]$, showing the non-existence of closed geodesics with one self-intersection. Here, $A(r)$ depends on the reversibility $r$ of the metric. This result is sharp in the following way: for any $e \gt 0$, we present $r$-reversible Finsler metrics with curvature in $[A(r)-e,1]$ with closed geodesics containing exactly one self-intersection. The main tools used come from the theory of pseudo-holomorphic curves in symplectizations of contact manifolds. This is a joint work with U. Hryniewicz (UFRJ).